Abstract
We classify the solutions to the equation (−Δ)m u = (2m − 1)!e 2mu on \({\mathbb{R}^{2m}}\) giving rise to a metric \({g=e^{2u}g_{\mathbb{R}^{2m}}}\) with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric \({e^{2u}g_{\mathbb{R}^{2m}}}\) at infinity, and we observe that the pull-back of this metric to S 2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.
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Martinazzi, L. Classification of solutions to the higher order Liouville’s equation on \({\mathbb{R}^{2m}}\) . Math. Z. 263, 307–329 (2009). https://doi.org/10.1007/s00209-008-0419-1
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DOI: https://doi.org/10.1007/s00209-008-0419-1